Rheological models for granular materials play an important role in the numerical simulation of dry dense snow avalanches. This article describes the application of a physically based model from the field of kinetic theory to snow avalanche simulations. The fundamental structure of the so-called extended kinetic theory is outlined and the decisive model behavior for avalanches is identified. A simplified relation, covering the basic features of the extended kinetic theory, is developed and implemented into an operational avalanche simulation software. To test the obtained friction relation, simulation results are compared to velocity and runout observations of avalanches, recorded from different field tests. As reference we utilize a classic phenomenological friction relation, which is commonly applied for hazard estimation. The quantitative comparison is based on the combination of normalized residuals of different observation variables in order to take into account the quality of the simulations in various regards. It is demonstrated that the extended kinetic theory provides a physically based explanation for the structure of phenomenological friction relations. The friction relation derived with the help of the extended kinetic theory shows advantages to the classic phenomenological friction, in particular when different events and various observation variables are investigated.

Within the past few decades several software tools for the
simulation of snow avalanches or, generally speaking, shallow granular flows
have been developed, such as SamosAT

Within this framework rheological models attract a significant portion of
attention. A widespread, classic phenomenological rheological model still
utilized in depth-averaged models is the Voellmy friction relation

The rheology of granular materials has been investigated in many scientific
works within the framework of three-dimensional continuum mechanics. An
important category of microrheological models, dealing with rapid granular
flows, is the kinetic theory

To implement the constitutive model into the depth-averaged dynamic model, several assumptions about the vertical structure of the flow are made, yielding a simplified friction relation. It is shown that the simplified expression is similar to classic friction relations. In a further step the obtained relation is compared to classic friction relations which are often applied in natural hazard estimation. For this task, back calculations of well-documented avalanches are conducted and minimal residuals in multiple observation variables, such as runout distance, affected area and velocity, are determined. It is shown that the relation obtained by kinetic theory allows to reduce the residuals for the presented events.

The governing equations of the Savage–Hutter model, extended for
entrainment, as implemented in SamosAT, for an incompressible, granular flow
over a one-dimensional terrain can be expressed as

The resistance of the material against its deformation is considered with the
second and third term on the right-hand side of Eq. (

The third term on the right-hand side of Eq. (

In this section we outline the fundamental structure of the extended kinetic
theory for simple shear conditions. Classic kinetic theories are based on the
statistical description of binary particle collisions, which are the
governing processes at low volume fractions. The extended kinetic
theory used here includes extensions

The extended kinetic theory as formulated by

Simple shear test: C denotes the center of gravity with an arrow showing the direction of its displacement during shearing.

The critical state theory was developed in soil mechanics, where the
deformation rates are very small and therefore no inertial forces occur. This
theory can be outlined with the help of the simple shear test with constant
normal stress

The shear stress at critical state is related to the applied normal stress:

The second constituent of the applied theory is the kinetic theory, which
provides relations for the normal and shear stress based on statistical
considerations of the collisions between particles in the volume

The main idea of the extended kinetic theory by

The simple shear setup, on which the presented model has been developed, is
shown in Fig.

Simple shear flow setup with linear velocity profile

Flows of granular material can display a large span of grain concentrations.
Microscopic mechanical processes and consequently the macroscopic behavior of
the material changes substantially with the concentration or solid fraction

Concentration and granular temperature are determined by the hydrodynamic
fields and their gradients. For the presented simple shear setup, these are
the normal stress along the transversal direction

To describe the whole range of flow configurations, multiple mechanical
processes, described by different theories, need to be taken into account

On the one hand, the critical state theory

On the other hand, the kinetic theory of granular gases describes the
granular material under the influence of high shear rates. The stresses in
the material are based on short contacts between particles, i.e.,
elastoplastic collisions. The following form of this theory was proposed by

The contribution from collisions to the total stresses is given as

Material parameters of the extended kinetic theory.

The total stresses can be expressed as sum of the quasi-static and the
collisional stresses, combining critical state theory and kinetic theory:

By introducing Eqs. (

Critical state surface in the

Stress ratio

Concentration

Contribution of enduring force chains: stress ratio

Contribution of collisions: stress ratio

Material parameters for snow are not available at the current stage. However,
they can be estimated from the results of back calculations of real-scale
avalanches, conducted in the second part of this paper. To qualitatively
highlight the most important features of the constitutive model, it is
analyzed for an idealized 1 mm quartz sand (

Figure

The granular temperature, which is not shown, is almost solely dependent on
the shear rate

The stress ratio

A separation of quasi-static and collisional stresses is shown in
Figs.

It is not feasible to implement the complete extended kinetic theory model in
an operational simulation tool. The main reason is that for certain
combinations of normal stress

The second approach is given as

The factors

Concentration

The factor

The constitutive model obtained by the granular kinetic theory results in a
relation depending on the shear rate, which does not explicitly appear in
depth-averaged models. However, the equilibrium of stresses at the bottom of
the avalanche requires that

Orientation of the coordinate system and stresses in the slope
parallel direction on an infinitesimal small control volume. Slope parallel
normal stresses (

Supposing that the avalanche has reached its steady state on a slope with
constant inclination

Velocity profile for an infinite avalanche in steady state on a
uniformly steep slope for the given rheology model (blue line). For
comparison, velocity measurements of a dry snow avalanche from a field test
in Vallée de la Sionne is shown (red filled circle is the mean velocity;
bars are
fluctuations)

The obtained velocity profile differs from the plug flow profile assumed by

Finally a relation between the depth-averaged velocity (Eq.

Expression (

The difference between the obtained relation and the friction relation of
Voellmy is the inverse quadratic dependency on the flow height. This leads to
a lower friction for larger flow heights and therefore larger avalanches.
This behavior is in line with observations, where the runout is often related
to the volume of the avalanche

To test the obtained friction relation, we employ a multivariate optimization
method, based on the work of

avalanche no. 103 from 10 February 1999 at the Vallée de la
Sionne (VdlS) test site with a deposition volume of approximately
500 000

avalanche from 17 April 1997 at the Ryggfonn (Rgf) test site
with a deposition volume of approximately 40 000

Especially for the VdlS avalanche, entrainment appears important because of
the high increase of volume during its descent. A simple approach for the
entrainment rate

In order to investigate the range of possible simulation results, the
friction parameter

To judge the quality of our simulations, they are compared with measurements of the following three observation variables.

The

The

The

This method does not require reference values like an acceptable error or a
measurement error. A possible drawback is that larger events have a bigger
impact on the results than smaller ones because of the larger absolute values
of velocity and runout. If this is not suitable for the respective problem,
one could also perform the normalization before combining the events and
therefore lay weight on different events equally. The combination of events
and measures leads to four possibilities to evaluate and compare model
performance with respect to different regards and events (compare
Figs.

to a single event with respect to a single observation variable
(

to a single event with respect to all investigated observation variables
(

to both events with respect to a single observation variable
(

to both events with respect to all investigated observation variables
(

The following section shows the evaluation of 1600 simulation runs. This
number results from two events, two friction models and 20 values for the
friction parameters

In Fig.

Outlines of the numerical simulations in comparison with the
documented affected area for the avalanche event in Ryggfonn

Velocity measurements compared with simulation outcomes for the
avalanche event in Ryggfonn

To gain a first insight concerning the validity of the back-calculated model
parameters for the two investigated friction relations, we determine the
optimal parameter set for the Rgf event and obtain

The runout distance represents a point in the avalanche path.
Simulations with high friction (high values for

The affected area is also a measure related to runout. However, it
provides an additional important information on the lateral extend and
spatial distribution of the avalanche. The correlation to runout is clearly
visible in Figs.

In case of the Rgf avalanche, it is observed that the agreement of documented
and simulated affected area is limited. This can be attributed to a large
amount of deposited snow in the runout, which is not considered in the
digital elevation model, leading to an upstream spreading of the avalanche
(see Fig.

For the VdlS event, the delineation of the documented affected area is
accompanied by high uncertainties due to the large powder cloud of this
avalanche. The applied documented affected area represents areas with clearly
visible snow depth variations (deposition and erosion) caused by the
avalanche

Another interesting detail can be observed in Fig.

The velocity along the radar path is visualized in
Fig.

In Fig.

The dynamic pressure can be calculated from velocity with Eq. (

Areas in the parameter space with relatively small residuals (less
than 10 % on the normalized scale) for the runout length (blue), the
affected area (red) and the velocity (yellow) for the different events and
friction models. The triangle in the respective color marks the simulation
with the smallest residual

Obtained residuals for all possible result evaluations. Each symbol
marks a set of parameters which can also be seen in the parameter space in
Figs.

Possible best-fit parameters can be obtained by analyzing the overlapping
areas in Fig.

The combined residual of velocity, runout length and affected area is shown
in Fig.

The parameters for the kinetic theory model, which yield the smallest
combined residual for both events (e.g., residual in velocity, runout and
affected area, white circles in Fig.

For comparison, practical guidelines propose the variation of Voellmy
friction parameters with different avalanche characteristics; e.g.,

Since a direct measurement is hardly possible, optimized values for the
parameters

The tangent of the critical friction angle

Stress ratio

From Fig.

This tendency increases with the number of observations combined.
Table

This improvement can be obtained with very little modification to current
models and simulation tools; i.e., no additional transport equation needs to
be solved. Convection and diffusion are neglected and a local equilibrium of
the granular temperature is assumed to get an analytical expression for the
shear stress. An additional improvement with a more accurate description of
the velocity profile is expected. A more realistic velocity profile should
also lead to different friction in head and tail of the avalanche like
proposed by

Overall, velocities predicted by the presented models can match the
observations quite well with an optimized set of parameters. However, this may
also be attributed to the considered entrainment process since the analysis
of similar friction approaches showed less agreement of the velocities,
disregarding entrainment

The evaluation of the affected area is accompanied by large uncertainties in the documentations. Therefore, assumptions about the quality of the model in this regard are limited.

Another problem of observations in the runout zone is the rising temperature
of the avalanche with its descent. The temperature increases because of
dissipating kinetic energy and entrainment of warm snow

For future works we suggest to include the energy conservation into the
dynamic flow model to track the evolution of the thermodynamic temperature.
Consequently, the particle diameter can be calculated (e.g., with the relation
proposed by

The possibility of a negative coefficient

The extended kinetic theory predicts a relation between classic flow
variables (e.g.,

Moreover, the transition point between dense flow regime and purely
collisional regime, where the density decreases rapidly is a meaningful
result for the simulation of powder snow avalanches. We found a simplified
empirical description for the transition point, when applying the microscopic
material parameters for snow:

In summary this paper highlights the application of a rheological model based on kinetic theory to depth-averaged snow avalanche simulations. To combine both frameworks we employed the commonly accepted assumption of a constant velocity profile along the avalanche and during its decent. The resulting relation shows similarities to classic friction relations. The employed comparison method allowed to evaluate the different basal friction models with respect to different observation variables. Here the residual sum of squares in combination with a normalization, such that values with different physical units and orders of magnitude can be combined, allowed the comparison of the presented friction relation to the widespread Voellmy friction relation. Utilizing the new relation shows some improvements, particularly when evaluating different observation variables and multiple events.

The underlying data are intellectual property of the BFW and its scientific partners and are not available to the public. For scientific collaboration and data usage, interested researchers are invited to get in contact with the authors.

We would like to express our gratitude to the Norwegian Geotechnical Institute (NGI; P. Gauer) and the WSL Institute for Snow and Avalanche research (SLF; B. Sovilla and P. Bartelt) for collaborative experiments at the Ryggfonn and Vallée de la Sionne test sites and also for access to the respective data and P. Bartelt for fruitful discussions. We thank M. Granig (WLV – Austrian Service for Torrent and Avalanche Control) and P. Sampl (AVL LIST GmbH) for their support and cooperation regarding SamosAT. Furthermore, the authors acknowledge the financial support by the OEAW project “beyond dense flow avalanches”. Edited by: P. Bartelt Reviewed by: two anonymous referees